United StatesDepartment ofAgriculture

Forest Service

NortheasternResearch Station

Research Paper NE-718

William B. Leak

Origin of SigmoidDiameter Distributions

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The Author

WILLIAM B. LEAK, a research forester with the Northeastern Research Station atDurham, New Hampshire, received B.S. and M.F. degrees from the State University

of New York College of Forestry. He joined the Station in 1956 and conducts

research on silviculture, site evaluation, and ecology of northern hardwoods.

Manuscript received for publication 24 September 2001

Abstract

Diameter distributions—numbers of trees over diameter at breast height (d.b.h.)—

were simulated over 20-years using six diameter-growth schedules, six mortality

trends, and three initial conditions. The purpose was to determine factors

responsible for the short-term development of the arithmetic rotated sigmoid form of

diameter distribution characterized by a plateau, near plateau, or bump in the mid-

diameter d.b.h. classes. A distinct rotated sigmoid developed when the diameter-

growth schedule dropped precipitously in the mid-diameter classes; this might reflect

a decadent overstory and thrifty understory. When the initial diameter distribution

was distinctly sigmoid, diameter-growth schedules characterized by slow growth in

the small-diameter classes tended to maintain sigmoid characteristics. A parabolic

growth trend with or without parabolic mortality, also produced moderate rotated

sigmoid characteristics. Under most other growth or mortality schedules, the 20-year

diameter distribution was J shaped. The results reinforce the concept that diameter

distributions tend to maintain or return to the J shape particularly with respect to

New England northern hardwoods.

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IntroductionF. DeLiocourt (1898) and H. Arthur Meyer (1952) wereright! In both managed and undisturbed forests, there isa pervasive tendency for the diameter distribution—number of trees over diameter at breast height (d.b.h.)—to approximate the reverse J shape. Many other fieldstudies have corroborated the utility of this general curve(e.g., Nyland 1998). Through simulation, I examinedthe effects of a variety of diameter-growth trends,mortality schedules, and initial conditions on the short-term development of the diameter distribution in NewEngland northern hardwoods. The reverse J shapepredominated in many (but not all) situations.

The primary purpose of this study was to determine thefactors that might lead to another form of diameterdistribution — the reverse or rotated sigmoid form. Forthis purpose, rotated sigmoid refers to a curve ofnumber of trees over d.b.h. that has an obvious plateau,near plateau, or bump in the mid-d.b.h. range.Logarithmic rotated sigmoid is a curve with a plateau,near plateau, or bump when the log of tree numbers isplotted over d.b.h. A reverse J shape describes a curve ofnumber of trees over d.b.h. that is steeply and steadilydeclining. Many distributions that appear J shaped plotas logarithmic rotated sigmoid. When a J-shape curveplots as a straight line on semi-log paper, thedistribution is called the negative exponential; this curvehas a constant quotient (Q) between numbers of trees insuccessive d.b.h. classes.

Goff and West (1975) suggested that the logarithmicrotated sigmoid was the natural steady-state distributionin small unmanaged hardwood-hemlock stands inWisconsin, and that it might result from low mortalityand rapid diameter growth of trees entering the maincrown canopy; some of their stands also would appearas rotated sigmoid (without logarithmictransformation). Lorimer and Frelich (1984) suggestedthat logarithmic rotated sigmoids were natural steady-state distributions in unmanaged Michigan hardwoods.They based their conclusion on long-term simulationsthat included parabolic diameter-growth and mortalityfunctions (rapid growth and low mortality in the midsizes). However, the initial distributions in their twoold-growth stands were rotated sigmoid (withoutlogarithmic transformation). Schmelz and Lindsey(1965) believed that the logarithmic rotated sigmoidtendency was a temporary stage in the recovery ofmidwestern old-growth hardwoods from an earlierdisturbance. Some of the economically optimal andsustainable distributions developed through simulations(Adams and Ek 1974; Gove and Fairweather 1992) havemoderate rotated sigmoid properties that would bemore pronounced with logarithmic transformation. Arecent study in the Lake States (Goodburn and Lorimer

1999) showed that old-growth northern hardwoodstands classified as balanced (based on areas of free-to-grow crowns) had logarithmic rotated sigmoiddistributions, whereas negative exponential distributionswere most common in selection stands classified asbalanced. Several northern hardwood stands in NewHampshire developed logarithmic rotated sigmoiddiameter distributions (though J shaped) following avariety of partial cuttings, whereas a nearby old-growthstand remained as a negative exponential distribution(Leak 1996); several stands that appeared J shapedexhibited a parabolic trend of log (number of trees) overd.b.h., which implies an increasing Q with increasingd.b.h., a form of distribution described some years ago(Leak 1964).

There are examples of intensively managed stands thatdeveloped rotated sigmoid (nonlogarithmic)distributions. On the Fernow Experimental Forest inWest Virginia, a hardwood stand harvested four timesover 24 years by removing trees throughout the d.b.h.range developed rotated sigmoid tendencies with aplateau in the mid-d.b.h. classes (Fig. 1). A comparablestand in which only sawlog-size trees were harvesteddeveloped a J-shape form (Trimble and Smith 1976). A30-acre northern hardwood stand on the BartlettExperimental Forest in New Hampshire developed adistinct bump in the mid-d.b.h. classes following threecuts over 48 years (Fig. 2).1

Thus, the literature conflicts somewhat as to the originand occurrence of various forms of diameterdistribution. It remains unclear whether sigmoid andother forms of diameter distribution are characteristic ofold-growth, disturbed stands or intensively managedstands. In addition, there is little information on how avariety of growth and mortality schedules and initialconditions influence the development and maintenanceof sigmoid vs. J-shape distributions. To further definethe influential factors particularly as related toconditions in New England, simulations were conductedto determine the effects of six diameter-growth trends,six mortality schedules, and three initial diameterdistributions on the development of diameterdistributions over 20 years. A wide range of growth andmortality patterns was included to produce pronouncedeffects, and because field studies continue to show thatgrowth and mortality trends are variable andunpredictable due to mixed species, suppression, siterelationships, insect/disease effects, and natural damage(e.g., ice storms).

1Leak, W. B.; Sendak, P. E. Changes in species, grade, andstructure over 48 years in a managed New Englandnorthern hardwood stand. In preparation.

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Figure 1.—Stand structure after 24 years of single-tree selection: a tract on theFernow Experimental Forest, West Virginia, where harvesting was conductedthroughout the range of d.b.h. classes (Trimble and Smith 1976).

Figure 2.—Stand structure after 48 years of single-tree selection: a tracton the Bartlett Experimental Forest, New Hampshire.1

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MethodsThe growth and mortality data were bounded by theresults of a comprehensive study of density and structurein a small sawtimber stand (somewhat even-aged withholdovers) on the Bartlett Experimental Forest(Solomon 1977) in New Hampshire. As a result, thefindings are especially applicable to conditions in NewEngland. Preliminary work with the simulation modelas well as long-term field experiments (e.g., Leak 1996)indicated that diameter distributions in New Englandhardwoods are resistant to perturbations. Accordingly,the study was designed to examine extreme conditionsand to detect visually distinct effects.

The six diameter-growth trends (Fig. 3) were:

Level.—Diameter growth was constant across all d.b.h.classes from 2 to 24 inches. This is fairly characteristic ofthe Solomon (1977) study in which poletimber andsawtimber grew at about the same rate. This trend tendsto be typical of fairly good sites in New England (Leak1983).

Decrease.—Diameter growth declined with d.b.h. Thistrend tends to be characteristic of mediocre sites in NewEngland where initial diameter growth is high but fallsoff rapidly with increasing d.b.h. (Leak 1983).

Increase.—Diameter growth increased with d.b.h. (Leakand Graber 1976). This trend has been observed in olderNew England hardwood stands in which the overstory istoo heavy or dense or the understory has been slow torespond — perhaps due to long periods of suppression.

In some instances, apparently where the largestoverstory trees are low in vigor, the trend is:

Parabolic.— Growth is slow in the largest and smallesttrees and greatest in the mid-d.b.h. classes. This trend issimilar to that tested by Lorimer and Frelich (1984) andpartially represents the theory proposed by Goff andWest (1975) for the development of sigmoiddistributions.

Stepdown.— Growth is rapid in the small classes andslow in the large ones. It is intended to represent anoverstory of low-vigor or diseased trees (e.g., a beechbark disease infestation) coupled with a vigorousunderstory, possibly of a different species. For example,data from Solomon’s (1977) study show thatpoletimber eastern hemlock (Tsuga canadensis) can growtwo to six times as fast as beech (Fagus grandifolia) orsugar maple (Acer saccharum) poletimber even underfairly dense stockings of 80 to 100 ft2 of basal area peracre.

Stepup.— Growth is slow in the small sizes and rapid inthe large ones. This trend might be typical of a fast-growing intolerant/intermediate overstory with anunderstory of slower growing tolerants.

The six mortality schedules (percent mortality) weredefined as for the growth trends (Fig. 4). Again,Solomon’s (1977) data were used as bounds. Theexception was the Parabolic trend in which the highmortality rates were in the large and small trees. Thistrend was similar to that tested by Lorimer and Frelich(1984). By contrast, Solomon (1977) found fairlyconstant mortality rates in both poletimber andsawtimber. Apart from these results, there are fewavailable data with which to corroborate the variousmortality schedules. However, the scenarios describedunder the growth trends apply generally to the mortality

Figure 3.—Annual d.b.h. growth trends used as simulation input.

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schedules, if it is assumed that slow growth implies highmortality and vice versa.

The three initial conditions (Fig. 5) represented: (1) atypical residual stand in uneven-aged northernhardwoods: 80 ft2 of basal area per acre (trees 2 inchesand larger in d.b.h.), 72 ft2 in trees 6 inches and larger, aQ of 1.5 and a maximum tree diameter of 24 inches; (2)an initial sigmoid distribution with a constant number

of trees in the 6- through 12-inch classes, 79 ft2 of basalarea, and 68 ft2 in trees 6 inches and larger in d.b.h. (thiscondition was designed to represent the situation wherepast disturbance or heavy marking in the small and largesizes had created sigmoid characteristics); and (3) aresidual stand from a diameter-limit cut with 80 ft2 peracre total, 69 ft2 in trees 6 inches and larger in d.b.h. andno trees above 14 inches. Only the Level growth/mortality schedules were applied to this initial condition.

Figure 4.—Annual percent mortality schedules used as simulation input.

Figure 5.—Initial diameter distributions: negative exponential, Q = 1.5, 80ft2 of basal area per acre; sigmoid distribution, 79 ft2 of basal area per acre;and diameter-limit, 80 ft 2 of basal area per acre.

D.b.h.

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The simulation model was a simple stand projectionsystem written in Visual Fortran.2 Beginning with aninitial stand, percent mortality was applied by d.b.h.class. Trees moved into the next d.b.h. class based on theproportion between d.b.h. growth and d.b.h. classwidth. Ingrowth was determined by applying a constantproportion (0.10) to the tree numbers in the 2-inchclass. The model was designed to incorporate the effectsof stand density on d.b.h. growth, mortality, andingrowth. However, for this study, the growth andmortality schedules in Figures 3 and 4 were applied over20 years. This short projection period represents a fairlytypical cutting cycle in northern hardwoods; using itavoids the inaccuracies inherent in long-term

projections, especially those caused by regenerationcycles. The results are presented in untransformednumbers of trees in the 6-inch and larger d.b.h. classsince the logarithmic form tends to exaggerate anysigmoid tendency, and because most managementdecisions and interpretations are based onuntransformed distributions. The simulations are voidof natural variability, so no statistical tests were applied.Interpretations of the results were based on visuallydistinct effects: the distribution was classified as arotated sigmoid if it exhibited a plateau, near plateau, orincline in the mid-d.b.h. classes.

ResultsWhen the mortality schedules (Fig. 4) were applied incombination with the Level growth schedule, most ofthe diameter distributions exhibited a J shape (orslightly sigmoid form) in 20 years regardless of theinitial stand condition (Figs. 6-7). The exception was theStepdown schedule which coupled with the sigmoidinitial condition produced a mid-diameter plateau

2The use of trade, firm, or corporation names in this paperis for the information and convenience of the reader. Suchuse does not constitute an official endorsement or approvalby the U.S. Department of Agriculture or Forest Service ofany product or service to the exclusion of others that maybe suitable.

Figure 6.—Simulated diameterdistributions after 20 years using(A) the Initial, Level, Increase, andStepdown mortality schedules, and(B) the Decrease, Parabolic, andStepup mortality schedules. Allsimulations using the Level growthtrend and beginning with thenegative exponential distribution.

A

B

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characteristic of the rotated sigmoid distribution.Decreasing trends in percent mortality (Stepdown,Decrease, and Parabolic schedules) resulted in curvesthat were less steep than those of other schedules. Thesigmoid initial condition resulted in diameterdistributions that are relatively steep in the lower d.b.h.classes (Fig. 7), that is, the Q between d.b.h. classes islarger in the smaller than in the larger classes. This is thetype of diameter distribution recognized by Hansen andNyland (1987), i.e., J shaped but with a larger Qrepresenting the smaller d.b.h. classes.

When the growth schedules (Fig. 3) were applied withthe Level mortality schedule, the diameter distributionsremained essentially J shaped when the initial conditionwas J shaped (Fig. 8). There were three exceptions. First,the Stepdown growth schedule produced a well-definedsigmoid diameter distribution; this results when treesmove into the mid-diameters faster than they move out.

As mentioned previously, this general form of d.b.h.growth might be caused by a somewhat decadentoverstory coupled with an understory influx of a morevigorous species. In fact, two long-term field studies inNew Hampshire that exhibited sigmoid tendencies overtime had a beech overstory with a developing hemlockunderstory (Leak 1996) or a hemlock understory with abubble of midtolerant species.1

The Stepup growth schedule also produced a deviantfrom the J-shape distribution in the form of a deficiencyin the 14-inch d.b.h. class (Fig. 8). Apparently, this wasthe result of trees moving out of this class faster thanthey move in. The Parabolic growth form also produceda slight plateau in the diameter distribution. Thispartially supports the results by Lorimer and Frelich(1984) as well as the suggestion of Goff and West(1975) that vigorous mid-diameter growth mightproduce a sigmoid distribution.

Figure 7.—Simulated diameterdistributions after 20 years using(A) the Initial, Level, Increase, andStepdown mortality schedules, and(B) the Decrease, Parabolic, andStepup mortality schedules. Allsimulations using the Level growthtrend and beginning with thesigmoid diameter distribution.

A

B

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When the initial distribution was sigmoid, scheduleswith slow d.b.h. growth in the small sizes (Parabolic,Stepup, Increase) responded slowly over the 20-yearsimulation period. This tended to maintain the sigmoidform (Fig. 9). The Stepdown trend produced a well-defined sigmoid distribution. Under the Decrease andLevel growth schedules, the J-shape form essentiallyreturned but with a steep (high Q) segment in thesmaller d.b.h. classes.

Simulations beginning with a J-shape distribution usinga mixture of decreasing growth and mortality ordecreasing growth and increasing mortality maintainedthe J shape (Fig. 10). The Parabolic combination (rapidgrowth and high mortality in the mid-diameters)produced a slight sigmoid tendency in the 20-yeardistribution.

Beginning with the residual stand from the diameter-limit cut, the Level growth and mortality schedulesresulted in a return to a fairly smooth J-shape (Fig. 11).

This result agrees with earlier findings on the BartlettForest where diameter-limit cuts through the 10-inch or13-inch d.b.h. classes developed J-shape curves,sometimes with moderate logarithmic sigmoidtendencies (Leak 1996).

DiscussionA simulation study cannot account for all of thebiological and historical factors that affect diameterdistributions, for example, historical cutting patterns,species mixtures, insect/disease infestations andregeneration cycles. However, several generalizations(backed by long-term field observations) can be madewith respect to New England northern hardwoods.

First, for a variety of growth and mortality schedules,there is a tendency to maintain or revert to a J-shapediameter distribution, sometimes with steeper segmentsin the lower d.b.h. range (Hansen and Nyland 1987).Beginning with a J-shape initial distribution, a distinct

Figure 8.—Simulated diameterdistributions after 20 years using (A)the Initial, Level, Increase, andStepdown growth schedules, and (B)the Decrease, Parabolic, and Stepupgrowth schedules. All simulationsusing the Level mortality scheduleand beginning with the negativeexponential distribution.

A

B

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sigmoid distribution resulted from a Stepdown growthschedule, which is characterized by a sharp drop ind.b.h. growth rates in the mid-diameter classes. Thisform could represent a decadent overstory with a morevigorous understory of perhaps a different species, e.g.,hemlock. A slightly sigmoid form was produced by aParabolic diameter-growth schedule (rapid mid-diameter growth) and to a lesser degree by Parabolicdiameter growth along with Parabolic mortality (lowmid-diameter mortality), as suggested by Goff and West(1975) and Lorimer and Frelich (1984).

Beginning with a distinctly sigmoidal initialdistribution, diameter-growth schedules with slowgrowth in the smaller d.b.h. classes tended to maintainsigmoid characteristics. Those with level d.b.h. growth orrapid growth in the smaller classes reverted to essentiallya J-shape structure over the 20-year simulation period.

The form of the mortality schedule had less effect on theresulting shape of the diameter distribution than thegrowth schedule.

With regard to management practices, it would seemthat a variety of marking/harvesting approaches can beused in northern hardwoods in New England withoutgreatly affecting the development of a diameterdistribution that resembles the J-shape or slightlysigmoid form. This observation agrees with long-termobservations in New England (Leak 1996). These formsgenerally are accepted as sustainable and productive, i.e.,they will support repeated harvesting over time. Oneexception that seems to significantly skew thedistribution toward the sigmoid form is heavy removalsin the smaller size classes, along with a low-vigor, slow-growing understory — possibly due to excessive stockingin the larger d.b.h. classes or long-term suppression ofthe smaller trees.

Are sigmoid distributions characteristic of old-growthstands, earlier disturbance, intensive management, oreconomically directed timber marking? The answerdepends primarily on how these scenarios affectdiameter-growth trends, initial stand conditions, and, toa lesser extent, mortality schedules.

Figure 9.—Simulated diameterdistributions after 20 years using (A)the Initial, Level, Increase, andStepdown growth schedules, and (B)the Decrease, Parabolic, and Stepupgrowth schedules. All simulationsusing the Level mortality scheduleand beginning with the sigmoiddiameter distribution.

A

B

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Figure 10.—Simulated diameter distributions after 20 years using growth andmortality combinations: growth Decrease/mortality Decrease and growthDecrease/mortality Increase (initial stand is negative exponential).

Figure 11.—Simulated diameter distribution after 20 years using Level growthand mortality (initial stand is a diameter limit).

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Literature CitedAdams, D. M.; Ek, A. R. 1974. Optimizing the

management of unevenaged forest stands. CanadianJournal of Forest Research. 4: 274-287.

DeLiocourt, F. 1898. De l’amenagement des Sapinieres.Bulletin De la Societe Forestiere de Franch-Conte etBelfort.

Goff, F. G.; West, D. 1975. Canopy-understoryinteraction effects on forest population structure.Forest Science. 21: 98-108.

Goodburn, J. M.; Lorimer, C. G. 1999. Populationstructure in old-growth and managed northernhardwoods: an examination of the balanceddiameter distribution concept. Forest Ecology andManagement. 118: 11-29.

Gove, J. H.; Fairweather, S. E. 1992. Optimizing themanagement of uneven-aged forest stands: astochastic approach. Forest Science. 38: 623-640.

Hansen, G. D.; Nyland, R. D. 1987. Effects of diameterdistribution on the growth of simulated uneven-aged sugar maple stands. Canadian Journal of ForestResearch. 17: 1-8.

Leak, W. B. 1996. Long-term structural change inuneven-aged northern hardwoods. Forest Science.42: 160-165.

Leak, W. B. 1983. Stocking, growth, and habitatrelations in New Hampshire hardwoods. Res. Pap.NE-523. Broomall, PA: U.S. Department ofAgriculture, Forest Service, Northeastern ForestExperiment Station. 11 p.

Leak, W. B. 1964. An expression of diameterdistribution for unbalanced, uneven-aged standsand forests. Forest Science. 10: 39-50.

Leak, W. B.; Graber, R. E. 1976. Seedling input, death,and growth in uneven-aged northern hardwoods.Canadian Journal of Forest Research. 6: 368-374.

Lorimer, C. G.; Frelich, L. E. 1984. A simulation ofequilibrium diameter distributions of sugar maple(Acer saccharum). Bulletin of the Torrey BotanicalClub 111: 193-199.

Meyer, H. A. 1952. Structure, growth, and drain inbalanced, uneven-aged forests. Journal of Forestry.52: 85-92.

Nyland, R. D. 1998. Selection system in northernhardwoods. Journal of Forestry. 96: 18-21.

Schmelz, D. V.; Lindsey, A. A. 1965. Size-class structureof old-growth forests in Indiana. Forest Science 11:258-264.

Solomon, D. S. 1977. The influence of stand densityand structure on growth of northern hardwoods inNew England. Res. Pap. NE-362. Upper Darby, PA:U.S. Department of Agriculture, Forest Service,Northeastern Forest Experiment Station. 13 p.

Trimble, G. R.; Smith, H. C. 1976. Stand structure andstocking control in Appalachian mixed hardwoods.Res. Pap. NE-340. Upper Darby, PA: U.S. Departmentof Agriculture, Forest Service, Northeastern ForestExperiment Station. 10 p.

Printed on Recycled Paper

Leak, William B. 2002. Origin of sigmoid diameter distributions. Res. Pap. NE-718.

Newtown Square, PA: U.S. Department of Agriculture, Forest Service, Northeastern

Research Station. 10 p.

Diameter distributions—numbers of trees over diameter at breast height (d.b.h.)—were

simulated over 20-years using six diameter-growth schedules, six mortality trends, and

three initial conditions. The purpose was to determine factors responsible for the short-

term development of the arithmetic rotated sigmoid form of diameter distribution

characterized by a plateau, near plateau, or bump in the mid-diameter d.b.h. classes.

A distinct rotated sigmoid developed when the diameter-growth schedule dropped

precipitously in the mid-diameter classes; this might reflect a decadent overstory and

thrifty understory. When the initial diameter distribution was distinctly sigmoid,

diameter-growth schedules characterized by slow growth in the small-diameter classes

tended to maintain sigmoid characteristics. A parabolic growth trend with or without

parabolic mortality, also produced moderate rotated sigmoid characteristics. Under

most other growth or mortality schedules, the 20-year diameter distribution was J

shaped. The results reinforce the concept that diameter distributions tend to maintain

or return to the J shape particularly with respect to New England northern hardwoods.

Headquarters of the Northeastern Research Station is in Newtown Square,

Pennsylvania. Field laboratories are maintained at:

Amherst, Massachusetts, in cooperation with the University of Massachusetts

Burlington, Vermont, in cooperation with the University of Vermont

Delaware, Ohio

Durham, New Hampshire, in cooperation with the University of New Hampshire

Hamden, Connecticut, in cooperation with Yale University

Morgantown, West Virginia, in cooperation with West Virginia University

Parsons, West Virginia

Princeton, West Virginia

Syracuse, New York, in cooperation with the State University of New York,

College of Environmental Sciences and Forestry at Syracuse University

Warren, Pennsylvania

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